Logics of √’qMV algebras Antonio Ledda Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks Università di Cagliari Siena, September 8 th 2008
Jan 01, 2016
Logics of √’qMV algebras
Antonio Ledda
Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks
Università di Cagliari
Siena, September 8th 2008
Some motivation
qMV algebras were introduced in an attempt
to provide a convenient abstraction of the
algebra over the set of all density
operators of the two-dimensional complex
Hilbert space, endowed with a suitable
stock of quantum gates.
The definition of qMV-algebra
Definition
Łukasiewicz’s axiom Smoothness axioms
qMV-algebras
Adding the square root of the negation
√’qMV algebras were introduced as term
expansions of quasi-MV algebras by an
operation of square root of the
negation.
Adding the square root of the negation
quasi-Wajsberg algebras
Term equivalence
Theorem
The standard Wajsberg algebra St
The algebra F[0,1]
The standard qW algebras S and D
Equationally defined preorder
An example of equationally defined preorder
Logics from equationally preordered classes
Remark
A logic from an equationally preordered variety
The quasi-Łukasiewicz logic qŁ
A remark
Summary of the logic results
A “logical” version of qMV
Term equivalences
Logics of qMV algebras (1)
Logics of qMV algebras (2)
Most logics in the previous schema look noteworthy under some respect:
Logics of qMV algebras (3)
1-cartesian algebras
Examples
Inclusion relationships
Placing our logics in the Leibniz hierarchy (1)
Well-behaved logics is regularly algebraisable and is its
equivalent quasivariety semantics;
is regularly algebraisable and is its equivalent quasivariety semantics;
(they are the 1-assertional logics of relatively
1-regular quasivarieties)
Placing our logics in the Leibniz hierarchy (2)
Ill-behaved logicsNone of the other logics is protoalgebraic:
: the Leibniz operator is not monotone on the deductive filters of F120;
: it is a sublogic of such;
: the Leibniz operator is not monotone on the deductive filters of ;
Placing our logics in the Leibniz hierarchy (2)
Placing our logics in the Frege hierarchy
Selfextensional logics
Non-selfextensional logics
Some notations
We use the following abbreviations:
The logics C and C1
The logics C and C1
A completeness result
The notion of (strong) implicative filter
Remark
In the definition of strong implicative filter conditions F2, F3, F4, F5 are redundant
A characterization of the deductive filters
Thank you for your attention!!